scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

scholarly journals The existence of small prime gaps in subsets of the integers

2015 ◽  
Vol 11 (03) ◽  
pp. 801-833 ◽  
Author(s):  
Jacques Benatar
Keyword(s):  
Arithmetic Progression ◽  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Small Constant ◽  
Prime Gaps

We consider the problem of finding small prime gaps in various sets [Formula: see text]. Following the work of Goldston–Pintz–Yıldırım, we will consider collections of natural numbers that are well-controlled in arithmetic progressions. Letting qn denote the nth prime in [Formula: see text], we will establish that for any small constant ϵ > 0, the set {qn | qn+1 - qn ≤ ϵ log n} constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao, we will also demonstrate that [Formula: see text] has bounded prime gaps. Specific examples, such as the case where [Formula: see text] is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets.

scholarly journals Monochromatic arithmetic progressions with large differences

1999 ◽  
Vol 60 (1) ◽  
pp. 21-35
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Constant Function ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

scholarly journals Improved bounds for arithmetic progressions in product sets

2015 ◽  
Vol 11 (08) ◽  
pp. 2295-2303 ◽  
Author(s):  
Dmitrii Zhelezov
Keyword(s):  
Lower Bound ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Complex Numbers ◽  
Natural Numbers ◽  
Multiplicative Constant ◽  
Product Sets

Let B be a set of natural numbers of size n. We prove that the length of the longest arithmetic progression contained in the product set B.B = {bb′|b, b′ ∈ B} cannot be greater than O(n log n) which matches the lower bound provided in an earlier paper up to a multiplicative constant. For sets of complex numbers, we improve the bound to Oϵ(n1 + ϵ) for arbitrary ϵ > 0 assuming the GRH.

scholarly journals On Rainbow Arithmetic Progressions

10.37236/1754 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Maria Axenovich ◽  
Dmitri Fon-Der-Flaass
Keyword(s):  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Natural Numbers

Consider natural numbers $\{1, \cdots, n\}$ colored in three colors. We prove that if each color appears on at least $(n+4)/6$ numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden's theorem proves the conjecture of Jungić et al.

Rainbow Arithmetic Progressions and Anti-Ramsey Results

2003 ◽  
Vol 12 (5-6) ◽  
pp. 599-620 ◽  
Author(s):  
V Jungic ◽  
J Licht ◽  
M Mahdian ◽  
J Nesetril ◽  
R Radoicic
Keyword(s):  
Arithmetic Progression ◽  
Ramsey Theory ◽  
Arithmetic Progressions ◽  
Natural Numbers ◽  
Colour Class ◽  
Ramsey Type ◽  
General Perspective

The van der Waerden theorem in Ramsey theory states that, for every k and t and sufficiently large N, every k-colouring of [N] contains a monochromatic arithmetic progression of length t. Motivated by this result, Radoičić conjectured that every equinumerous 3-colouring of [3n] contains a 3-term rainbow arithmetic progression, i.e., an arithmetic progression whose terms are coloured with distinct colours. In this paper, we prove that every 3-colouring of the set of natural numbers for which each colour class has density more than 1/6, contains a 3-term rainbow arithmetic progression. We also prove similar results for colourings of . Finally, we give a general perspective on other anti-Ramsey-type problems that can be considered.

POWERS FROM PRODUCTS OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSION

2006 ◽  
Vol 92 (2) ◽  
pp. 273-306 ◽  
Author(s):  
M. A. BENNETT ◽  
N. BRUIN ◽  
K. GYÖRY ◽  
L. HAJDU
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Diophantine Equations ◽  
Galois Representations ◽  
Rational Points ◽  
Arithmetic Progressions ◽  
Superelliptic Curves

We show that if $k$ is a positive integer, then there are, under certain technical hypotheses, only finitely many coprime positive $k$-term arithmetic progressions whose product is a perfect power. If $4 \leq k \leq 11$, we obtain the more precise conclusion that there are, in fact, no such progressions. Our proofs exploit the modularity of Galois representations corresponding to certain Frey curves, together with a variety of results, classical and modern, on solvability of ternary Diophantine equations. As a straightforward corollary of our work, we sharpen and generalize a theorem of Sander on rational points on superelliptic curves.

scholarly journals Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 247
Author(s):  
Kai An Sim ◽  
Kok Bin Wong
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Minimum Number ◽  
Necessary And Sufficient

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

On the Set of Common Differences in van der Waerden’s Theorem on Arithmetic Progressions

1999 ◽  
Vol 42 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Tom C. Brown ◽  
Ronald L. Graham ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Infinite Number ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Arithmetic Progressions ◽  
Large Set ◽  
Open Questions ◽  
The Family ◽  
Positive Integers

AbstractAnalogues of van derWaerden’s theorem on arithmetic progressions are considered where the family of all arithmetic progressions, AP, is replaced by some subfamily of AP. Specifically, we want to know for which sets A, of positive integers, the following statement holds: for all positive integers r and k, there exists a positive integer n = w′(k, r) such that for every r-coloring of [1, n] there exists a monochromatic k-term arithmetic progression whose common difference belongs to A. We will call any subset of the positive integers that has the above property large. A set having this property for a specific fixed r will be called r-large. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set {an : n = 1, 2,…} can have . Sufficient conditions for a set to be large are also given. We show that any set containing n-cubes for arbitrarily large n, is a large set. Results involving the connection between the notions of “large” and “2-large” are given. Several open questions and a conjecture are presented.

scholarly journals The least common multiple of consecutive arithmetic progression terms

2011 ◽  
Vol 54 (2) ◽  
pp. 431-441 ◽  
Author(s):  
Shaofang Hong ◽  
Guoyou Qian
Keyword(s):  
Positive Integer ◽  
Periodic Function ◽  
Arithmetic Progression ◽  
Arithmetic Function ◽  
Arithmetic Progressions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Image Position ◽  
Positive Integers

AbstractLet k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetic function gk,a,b for any positive integer n byIf we let a = 1 and b = 0, then gk,a,b becomes the arithmetic function that was previously introduced by Farhi. Farhi proved that gk,1,0 is periodic and that k! is a period. Hong and Yang improved Farhi's period k! to lcm(1, 2, … , k) and conjectured that (lcm(1, 2, … , k, k + 1))/(k + 1) divides the smallest period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the following interesting question: is gk,a,b periodic? If so, what is the smallest period of gk,a,b? We first show that the arithmetic function gk,a,b is periodic. Subsequently, we provide detailed p-adic analysis of the periodic function gk,a,b. Finally, we determine the smallest period of gk,a,b. Our result extends the Farhi–Kane Theorem from the set of positive integers to general arithmetic progressions.

scholarly journals A New Method to Construct Lower Bounds for Van der Waerden Numbers

10.37236/925 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
P. R. Herwig ◽  
M.J.H. Heule ◽  
P. M. Van Lambalgen ◽  
H. Van Maaren
Keyword(s):  
Lower Bounds ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
New Method ◽  
Natural Numbers

We present the Cyclic Zipper Method, a procedure to construct lower bounds for Van der Waerden numbers. Using this method we improved seven lower bounds. For natural numbers $r$, $k$ and $n$ a Van der Waerden certificate $W(r,k,n)$ is a partition of $\{1, \ldots, n\}$ into $r$ subsets, such that none of them contains an arithmetic progression of length $k$ (or larger). Van der Waerden showed that given $r$ and $k$, a smallest $n$ exists - the Van der Waerden number $W(r,k)$ - for which no certificate $W(r,k,n)$ exists. In this paper we investigate Van der Waerden certificates which have certain symmetrical and repetitive properties. Surprisingly, it shows that many Van der Waerden certificates, which must avoid repetitions in terms of arithmetic progressions, reveal strong regularities with respect to several other criteria. The Cyclic Zipper Method exploits these regularities. To illustrate these regularities, two techniques are introduced to visualize certificates.

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